Robert Waldmann and Tilman Tacke say it might not be. Holding constant levels of GDP per capita and the public health spending share of GDP, increases in private health spending are associated with higher--not lower--infant mortality:
Angry Bear: Public and Private Health Care Spending and Infant Mortality in 71 countries: We don't know if someone else has noticed this amazing fact: in a cross country regression, the ratio of public health care spending to GDP is negatively correlated with the infant mortality rate as one would expect, but the share of private health care spending in GDP is positively correlated. In a simple regression with including only log per capita GDP (corrected for PPP) as an additional explanatory variable, both coefficients have large t-statistics.
The positive coefficient on private health care spending becomes insignificant when other variables are included, but it does not become negative. The result is not due to the USA, which is an extreme outlier in private health care spending over GDP.
The result is not simply due to a correlation between high public spending and low income inequality as it holds when log per capita GDP is replaced by log per capita income of each of the lower 4 quintiles (this is our original regression hence the low number of countries in the sample):